A bicyclist is riding on a path modeled by the function
f(x) = 0.04(8x − x2),
where x and f(x) are measured in miles. Find the rate of change of elevation at x = 1.
f'(x) is the rate of change.
So, assuming f(x) is the elevation (which you don't actually state),
f'(x) = 0.04(8-2x)
so just plug in x=1
Note: apparently the units are miles, so f'(x) is in mi/mi.
What does that mean?
To find the rate of change of elevation at x = 1, we need to calculate the derivative of the function f(x) = 0.04(8x - x^2) and evaluate it at x = 1.
Step 1: Calculate the derivative of f(x)
To find the derivative, apply the power rule and constant multiple rule. The derivative of the function is given by:
f'(x) = 0.04(8 - 2x)
Step 2: Evaluate the derivative at x = 1
Substitute x = 1 into the derivative to find the rate of change of elevation at x = 1:
f'(1) = 0.04(8 - 2(1))
Simplifying,
f'(1) = 0.04(8 - 2)
f'(1) = 0.04(6)
f'(1) = 0.24
The rate of change of elevation at x = 1 is 0.24 miles per mile.
To find the rate of change of elevation at x = 1, we need to calculate the derivative of the function f(x) and evaluate it at x = 1. The derivative will give us the instantaneous rate of change of elevation at that point.
Let's start by finding the derivative of the function f(x). The formula for finding the derivative of a quadratic function is:
f'(x) = 2ax - b
where a and b are the coefficients of the quadratic equation. In this case, a = -0.04 and b = 0.
Now, let's plug in the values of a and b into the derivative formula:
f'(x) = 2(-0.04)x - 0
= -0.08x
To find the rate of change of elevation at x = 1, we substitute x = 1 into the derivative:
f'(1) = -0.08(1)
= -0.08
Therefore, the rate of change of elevation at x = 1 is -0.08. This means that for every mile the bicyclist travels, the elevation decreases by 0.08 miles.